MP3(aka MPEG1-LayerⅢ)の要素技術

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MP3 (aka MPEG1-LayerIII)ͷ ཁૉٕज़ aikiriao 2024.3 1 / 71

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͋Β͢͡ 1. MP3 ֓ཁ MPEG/Audio ͷྺ࢙ ίʔσοΫߏ଄ 2. ϋΠϒϦουϑΟϧλόϯΫ ϑΟϧλόϯΫ MDCT (Modiﬁed Discrete Cosine Transform) ΤΠϦΞε࡟ݮόλϑϥΠԋࢉ 3. ྔࢠԽ 4. ූ߸Խ 5. ௌ֮৺ཧϞσϧ II 6. dist10 Τϯίʔμͷ֎෦ɾ಺෦ϧʔϓ 2 / 71

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1. MP3 ֓ཁ MPEG/Audio ͷྺ࢙ ίʔσοΫߏ଄ 2. ϋΠϒϦουϑΟϧλόϯΫ ϑΟϧλόϯΫ MDCT (Modiﬁed Discrete Cosine Transform) ΤΠϦΞε࡟ݮόλϑϥΠԋࢉ 3. ྔࢠԽ 4. ූ߸Խ 5. ௌ֮৺ཧϞσϧ II 6. dist10 Τϯίʔμͷ֎෦ɾ಺෦ϧʔϓ 3 / 71

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MPEG/Audioͷྺ࢙ MPEG/Audio ͷܥේɽ[1] ΑΓҾ༻ 4 / 71

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MPEG/Audioͷྺ࢙ MPEG/Audio ͷؔ܎ɽ[1] ΑΓҾ༻ 5 / 71

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MPEG1ͷཁૉٕज़ ▶ ϨΠϠ I,II,III ͷॱ ʹѹॖ཰޲্ ▶ ϨΠϠ I,II ͸αϒ όϯυූ߸Խ͕ ϝΠϯ ▶ ௌ֮৺ཧϞσϧ ͸ I,II ͱ III Ͱҟ ͳΔ [1] ΑΓҾ༻ 6 / 71

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MP3ͷΤϯίʔμߏ଄ ϋΠϒϦουϑΟϧλόϯΫ 32 όϯυͷϑΟϧλό ϯΫͷޙɼ18 ఺ͷ MDCT → 576 ఺ͷεϖΫ τϧΛܭࢉ ྔࢠԽ ྟքଳҬɾϚεΩϯάͷ৘ใΛݩʹεϖΫ τϧΛྔࢠԽ ූ߸Խ ௿ҬΛਫ਼ີɾߴҬΛߥ͘ූ߸Խ 7 / 71

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MP3ͷσίʔμߏ଄ Τϯίʔμͷٯͷૢ࡞ 8 / 71

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1. MP3 ֓ཁ MPEG/Audio ͷྺ࢙ ίʔσοΫߏ଄ 2. ϋΠϒϦουϑΟϧλόϯΫ ϑΟϧλόϯΫ MDCT (Modiﬁed Discrete Cosine Transform) ΤΠϦΞε࡟ݮόλϑϥΠԋࢉ 3. ྔࢠԽ 4. ූ߸Խ 5. ௌ֮৺ཧϞσϧ II 6. dist10 Τϯίʔμͷ֎෦ɾ಺෦ϧʔϓ 9 / 71

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M ෼ׂϑΟϧλόϯΫ[2] ▶ ৴߸Λ M ݸͷଳҬʹ෼ׂ ▶ M ݸͷ෼ੳϑΟϧλ hk ɾ߹੒ϑΟϧλ fk Λ࢖༻ 10 / 71

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ίαΠϯมௐϑΟϧλόϯΫ[2, 3] ίαΠϯมௐϑΟϧλόϯΫ 1 ͭͷ࣮܎਺ɾ௚ઢҐ૬ϓϩτλΠϓϑΟϧλ p0 [n] ͔ Βɼ෼ੳϑΟϧλ hk ͱ߹੒ϑΟϧλ fk Λ࣍Ͱઃఆɿ hk [n] = 2p0 [n] cos π M k + 1 2 n − L − 1 2 + θk (1) fk [n] = hk [L − 1 − n] (2) Mɿ෼ׂଳҬ਺ɼLɿλοϓ௕ɼθk = (−1)k π 4 ৄࡉ͸ิ଍ 1 અʹهࡌ 11 / 71

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MP3ͷϑΟϧλόϯΫ M = 32, L = 33 ͱͨ͠ίαΠϯมௐϑΟϧλόϯΫʹ ͍͕ۙҟͳΔʂʢθk ͕ͳ͍ʂʣ hk [n] = p0 [n] cos π 32 k + 1 2 (n − 16) (3) fk [n] = 32p0 [n] cos π 32 k + 1 2 (n + 16) (4) p0 [n] = −Cn ⌊n/64⌋ ͕ۮ਺ Cn ⌊n/64⌋ ͕ح਺ (5) Cn (n = 0, ..., 511) ͸ن֨Ͱઃఆ 12 / 71

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ϑΟϧλόϯΫͷಛੑ 0 100 200 300 400 500 index 0.005 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 amplitude MP3 Encoder prototype filter coefficients ▶ p0 [n] ͷܗঢ়ɽରশʢ= ௚ઢҐ૬ಛੑΛ΋ͭʣ ɽ 13 / 71

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ϑΟϧλόϯΫͷप೾਺ಛੑ όϯΫ k = 0, ..., 15 ͷप೾਺ಛੑ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 normalized frequency 200 180 160 140 120 100 80 60 amplitude Bank 0 Bank 1 Bank 2 Bank 3 Bank 4 Bank 5 Bank 6 Bank 7 Bank 8 Bank 9 Bank 10 Bank 11 Bank 12 Bank 13 Bank 14 Bank 15 14 / 71

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ϑΟϧλόϯΫͷप೾਺ಛੑ όϯΫ k = 16, ..., 31 ͷप೾਺ಛੑ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 normalized frequency 200 180 160 140 120 100 80 60 amplitude Bank 16 Bank 17 Bank 18 Bank 19 Bank 20 Bank 21 Bank 22 Bank 23 Bank 24 Bank 25 Bank 26 Bank 27 Bank 28 Bank 29 Bank 30 Bank 31 15 / 71

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ϑΟϧλόϯΫͷ࣮૷ ϓϩάϥϜͰ͸ɼೖྗ x[t] ͔Βόϯυ k ͷग़ྗ yk [t] Λ yk [t] = 63 s=0 tk,s 7 u=0 x[t − s − 64u]Cs+64u tk,s := cos π 32 k + 1 2 (s − 16) Ͱܭࢉɽ͜ͷ͕ࣜ FIR ϑΟϧλग़ྗܭࢉࣜ yk [t] = 511 n=0 x[t − n]hk [n] = 511 n=0 x[t − n]p0 [n]tk,n (6) ͔Βಋ͔ΕΔ͜ͱΛࣔ͢ɽ 16 / 71

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ϑΟϧλόϯΫͷ࣮૷ (6) ࣜΛมܗ͍ͯ͘͠ͱɼ yk[t] = 511 n=0 x[t − n]p0[n]tk,n = 7 u=0 63 s=0 x[t − s − 64u]p0[s + 64u]tk,s+64u = 7 u=0 63 s=0 x[t − s − 64u](−1)uCs+64utk,s+64u (7) ͜͜Ͱɼ tk,s+64u = cos π 32 k + 1 2 (s + 64u − 16) = cos π 32 k + 1 2 (s − 16) + π (2k + 1) u = cos π 32 k + 1 2 (s − 16) cos [π(2k + 1)u] − sin π 32 k + 1 2 (s − 16) sin [π(2k + 1)u] = (−1)utk,s ͔ͩΒɼ͜ΕΛࣜ (7) ʹ୅ೖ͢Ε͹ɼ yk[t] = 7 u=0 63 s=0 x[t − s − 64u]Cs+64utk,s = 63 s=0 tk,s 7 u=0 x[t − s − 64u]Cs+64u ϓϩάϥϜͷܭࢉ͕ࣜಋ͔Εͨɽ 17 / 71

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ϑΟϧλόϯΫ͸׬શ࠶ߏ੒͔ʁ ▶ Cn ͷಋग़ํ๏͕ෆ໌ɽݫີʹ׬શ࠶ߏ੒ੑΛࣔ ͤͳ͍ ▶ ࠶ߏ੒৴߸ ˆ x[n] ͕ೖྗ৴߸ͷ஗Ԇ+ఆ਺ഒʹͳΔ ͔؍࡯                      yk [n] = 511 i=0 hk [i]x[n − i] όϯΫ k ͷ෼ੳϑΟϧλग़ྗ zk [n] = 511 i=0 fk [i]yk [n − i] όϯΫ k ͷ߹੒ϑΟϧλग़ྗ ˆ x[n] = 31 k=0 zk [n] ࠶ߏ੒৴߸ 18 / 71

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ϑΟϧλόϯΫ͸׬શ࠶ߏ੒͔ʁ ˆ x[n] = 31 k=0 zk [n] = 31 k=0 511 i=0 fk [i]yk [n − i] = 31 k=0 511 i=0 fk [i] 511 j=0 hk [j]x[n − i − j] = 31 k=0 511 i=0 511 j=0 fk [i]hk [j]x[n − i − j] = 31 k=0 1022 m=0 min{511,m} i=max{0,m−511} fk [i]hk [m − i]x[n − m] (m := i + j) = 1022 m=0 x[n − m] min{511,m} i=max{0,m−511} 31 k=0 fk [i]hk [m − i] =g[m] (8) 19 / 71

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ϑΟϧλόϯΫ͸׬શ࠶ߏ੒͔ʁ ▶ g[m] ͷάϥϑʢӈਤʣ ▶ g[m] ≈ 32δm,512 ͩ ͔Βɼ ˆ x[n] ≈ 32x[n − 512] ۙࣅతʹ׬શ࠶ߏ੒ 0 128 256 384 512 640 768 896 1024 0 5 10 15 20 25 30 amplitude 0 128 256 384 512 640 768 896 1024 samples 250 200 150 100 50 0 50 abs amplitude (dB) 20 / 71

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MP3ͷMDCT (Modiﬁed DCT) ▶ αϒόϯυϑΟϧλͰ 32 ଳҬʹ෼ׂͨ͠৴߸ʹ ର͠ɼ18 ఺ MDCT Λ࣮ߦ ▶ ग़ྗɿ32 × 18 = 576 ఺ͷεϖΫτϧσʔλ ▶ MDCT ͷલɾIMDCT ͷޙͰ૭ؔ਺Λద༻ ▶ MP3 Ͱ͸ 4 छྨͷ૭ؔ਺Λ࢖༻ ▶ ૭ؔ਺͸׬શ࠶ߏ੒৚݅Λຬͨ͢ 21 / 71

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MDCTͱIMDCT ೖྗ৴߸Λ x[n]ɼ࠶ߏ੒৴߸Λ y[n] ͱͯ͠ MDCT (Modiﬁed Discrete Cosine Transform) Xk = 2N−1 n=0 x[n] cos π N k + 1 2 n + 1 2 + N 2 (9) IMDCT (Inverse MDCT) y[n] = 2 N N−1 k=0 Xk cos π N k + 1 2 n + 1 2 + N 2 (10) ▶ ࣌ؒྖҬ͸ 2N ఺ɼप೾਺ྖҬ͸ N ఺ͷม׵ ʢk = 0, ..., N − 1ʣ 22 / 71

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MDCTͱ׬શ࠶ߏ੒৚݅ (10) ࣜʹ (9) ࣜΛ୅ೖͯ͠੔ཧ͢Δͱ y[n] = x[n] − x[N − 1 − n] (n = 0, ..., N − 1) x[n] + x[3N − 1 − n] (n = N, ..., 2N − 1) (11) ͱͳΔʢূ໌͸ิ଍ʣ ɽ ϋʔϑΦʔόʔϥοϓͰॲཧ͢Δͱ͖ɼx[n], y[n] ʹ͏ ·͘૭ؔ਺Λద༻͢Δͱ׬શ࠶ߏ੒ʹͰ͖Δɽ ͦͷ৚ ݅͸ʁ 23 / 71

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ϋʔϑΦʔόʔϥοϓΞυͷखॱ Τϯίʔυ σίʔυ 24 / 71

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MDCTͱ׬શ࠶ߏ੒৚݅ Princen–Bradley ৚݅ʢ׬શ࠶ߏ੒৚݅ʣ[4] ௕͞ 2N ͷ෼ੳ૭Λ wa ɼ߹੒૭Λ ws ͱͨ͠ͱ͖ɼ wa [n]ws [n] + wa [n + N]ws [n + N] = 1 (12) wa [n + N]ws [2N − 1 − n] = wa [n]ws [N − 1 − n] (13) n = 0, ..., N − 1 ͳΒ͹ɼMDCTɾIMDCT ʹΑΔϋʔϑΦʔόʔϥοϓ Ξυ͸׬શ࠶ߏ੒ 25 / 71

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Princen–Bradley৚݅ͷಋग़ m ϑϨʔϜ໨ͷ n ࣌ࠁͷೖྗ xm [n] ͸ɼϑϨʔϜ͋ͨ Γ N αϯϓϧͰεϥΠυ͓ͯ͠Γ xm [n] = xm−1 [n + N] (14) ͕੒Γཱͭͱ͢Δɽ૭͔͚ͨ͠৴߸ gm [n] Λ gm [n] := wa [n]xm [n] (n = 0, ..., 2N − 1) (15) ͱॻ͘ɽgm [n] Λ MDCTɾIMDCT ͯ͠࠶ߏ੒ͨ͠৴߸ zm [n] ͸ɼ(11) ࣜΑΓɼ zm [n] = gm [n] − gm [N − 1 − n] (n = 0, ..., N − 1) gm [n] + gm [3N − 1 − n] (n = N, ..., 2N − 1) (16) 26 / 71

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Princen–Bradley৚݅ͷಋग़ ϋʔϑΦʔόʔϥοϓΞυͨ݁͠ՌΛ ˆ xm [n] ͱॻ͘ͱɼ (16) ࣜΑΓɼ ˆ xm [n] = ws [n]zm [n] + ws [n + N]zm−1 [n + N] = ws [n](gm [n] − gm [N − 1 − n]) + ws [n + N](gm−1 [n + N] + gm−1 [3N − 1 − (n + N)]) = ws [n](wa [n]xm [n] − wa [N − 1 − n]xm [N − 1 − n]) + ws [n + N](wa [n + N]xm−1 [n + N] + wa [2N − 1 − n]xm−1 [2N − 1 − n]) = ws [n](wa [n]xm [n] − wa [N − 1 − n]xm [N − 1 − n]) + ws [n + N](wa [n + N]xm [n] + wa [2N − 1 − n]xm [N − 1 − n]) = xm [n](wa [n]ws [n] + wa [n + N]ws [n + N]) + xm [N − 1 − n](wa [n + N]ws [2N − 1 − n] − wa [n]ws [N − 1 − n]) ͜ͷ݁ՌΛ xm [n] = ˆ xm [n] ͱͯ྆͠ลൺֱ͢Δ͜ͱͰ ৚͕݅ಘΒΕΔ 27 / 71

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MP3ͱPrincen–Bradley৚݅ Princen–Bradley ৚݅ʢ෼ੳ૭ͱ߹੒૭͕ಉҰʣ ෼ੳɾ߹੒૭͕ಉ͡ w[n] = wa [n] = ws [n] ͱ͖ɼ w[n]2 + w[n + N]2 = 1 (17) w[n + N]w[2N − 1 − n] = w[n]w[N − 1 − n] (18) n = 0, ..., N − 1 ͱ͘ʹ૭ؔ਺͕ରশ w[n] = w[2N − 1 − n] Ͱ͋Ε͹ɼ w[n + N]w[2N − 1 − n] = w[N − 1 − n]w[2N − 1 − n] = w[N − 1 − n]w[n] ͱͳΓࣜ (18) ͕ຬͨ͞ΕΔ ∗1 ∗1ٯʹɼࣜ (18) Λຬͨͯ͠΋ରশͱ͸ݶΒͳ͍ɽw[n] = w[n+N]w[2N−1−n] w[N−1−n] ͕ͩɼҰൠʹ w[n + N] ̸= w[N − 1 − n] ͔ͩΒ w[n] ̸= w[2N − 1 − n]ɽ 28 / 71

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MP3ͱMDCT – 4छྨͷ૭ؔ਺ छྨ ఆٛ long w[n] = sin π 36 n + 1 2 (n = 0, ..., 35) short w[n] = sin π 12 n − 6k + 1 2 (k = 1, 2, 3, n = 6k, ..., 6k + 11) start w[n] =        sin π 36 n + 1 2 (n = 0, ..., 17) 1 (n = 18, ..., 23) sin π 12 n − 18 + 1 2 (n = 24, ..., 29) 0 (n = 30, ..., 35) stop w[n] =        0 (n = 0, ..., 5) sin π 12 n − 6 + 1 2 (n = 6, ..., 11) 1 (n = 12, ..., 17) sin π 36 n + 1 2 (n = 18, ..., 35) ▶ short ͷ૭௕͸ 12ɼͦΕҎ֎͸ 36 29 / 71

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MP3ͱMDCT – 4छྨͷ૭ؔ਺ 0 5 10 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 1.0 long 0 5 10 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 1.0 start 0 5 10 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 1.0 stop 0 5 10 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 1.0 short 30 / 71

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MP3ͱMDCT – ૭ؔ਺ͷঢ়ଶભҠ ঢ়ଶભҠ͸ௌ֮৺ཧϞσϧʹΑΓܾఆ 31 / 71

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MP3ͱPrincen–Bradley৚݅ αΠϯ૭ w[n] = sin π 2N n + 1 2 (n = 0, ..., 2N − 1) (19) ͸ Princen–Bradley ৚݅Λຬͨ͢ɽ ʢূ໌ʣࣜ (17) ͸ɿ w[n]2 + w[n + N]2 = sin2 π 2N n + 1 2 + sin2 π 2N n + N + 1 2 = sin2 π 2N n + 1 2 + cos2 π 2N n + 1 2 = 1 ࣜ (18) ͸ɼαΠϯ૭͕ରশͰ͋Δ͜ͱΑΓࣔ͞ΕΔɿ w[2N − 1 − n] = sin π 2N 2N − 1 − n + 1 2 = sin π + π 2N −n − 1 2 = sin − π 2N n + 1 2 + π = sin π 2N n + 1 2 = w[n] 32 / 71

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MP3ͱPrincen–Bradley৚݅ ▶ long, short ૭͸αΠϯ૭ͦͷ΋ͷͳͷͰ׬શ࠶ ߏ੒ ▶ ঢ়ଶભҠ࣌ʹ׬શ࠶ߏ੒ʹͳΔ͔ʁ 1. long → start 2. stop → longɿ1. ͷରশέʔε 3. start → short 4. short → stopɿ3. ͷରশέʔε 1. ͱ 3. ͚ͩ֬ೝ 33 / 71

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MP3ͱPrincen–Bradley৚݅ 1. long → start ▶ ϋʔϑΦʔόʔ ϥοϓΞυ͢Δ۠ ؒͰαΠϯ૭ʹ ͳ͍ͬͯΔͨΊɼ׬ શ࠶ߏ੒ 0 5 10 15 20 25 30 35 40 45 50 55 0.0 0.2 0.4 0.6 0.8 1.0 long to start window long (first-half) long (last-half) start (first-half) start (last-half) 34 / 71

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MP3ͱPrincen–Bradley৚݅ 3. start → short ▶ n = 0, ..., 5ɿstart ૭ ͕ 1, short ૭͕ 0 ͳ ͷͰ׬શ࠶ߏ੒ ▶ n = 6, ..., 11ɿαΠ ϯ૭ʹͳ͍ͬͯΔ ͨΊ׬શ࠶ߏ੒ ▶ n = 12, ..., 17ɿstart ૭͕ 0, short ૭Ͳ͏ ͠ͰαΠϯ૭ʹ ͳ͍ͬͯΔͨΊ׬ શ࠶ߏ੒ 0 5 10 15 20 25 30 35 40 45 50 55 0.0 0.2 0.4 0.6 0.8 1.0 start to short window start (first-half) start (last-half) short (first-half) short (last-half) 35 / 71

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ΤΠϦΞε࡟ݮόλϑϥΠԋࢉ ▶ ϑΟϧλόϯ Ϋ͸ɼྡ઀ό ϯΫͷप೾਺ ੒෼ʢΤΠϦ Ξεʣ͕ࠞೖ ▶ ྡ઀όϯΫͷ εϖΫτϧΛ ࢖͍ΤΠϦΞ ε࡟ݮʢ[5]ʣ 0 50 100 150 200 250 frequency bin 160 140 120 100 80 60 amplitude (dB) Aliasing reduction butterfly effect (no butterfly) 36 / 71

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ΤΠϦΞε࡟ݮόλϑϥΠԋࢉ ▶ εϖΫτϧΛ Xk ͱͨ͠ͱ͖ɼ X18k−i ← csi X18k−i − cai X18k+i+1 (20) X18k+i+1 ← cai X18k−i + csi X18k+i+1 (21) k = 1, ..., 31, i = 0, ..., 7 ▶ ܎਺ csi , cai ͷఆٛ csi := 1 1 + c2 i , cai := ci 1 + c2 i (22) ci (i = 0, ...7) ͸ن֨Ͱઃఆ [6] ΑΓҾ༻ 37 / 71

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ΤΠϦΞε࡟ݮόλϑϥΠԋࢉ ▶ 14-17 όϯυ ͷप೾਺ಛੑ ൺֱ ▶ όλϑϥΠԋ ࢉʹΑΓɼྡ ઀όϯΫͱަ ࠩ͢Δৼ෯ ͕-2dB ΄ͲԼ ʹҠಈʢվળʣ 105 110 115 120 125 130 135 140 145 150 frequency bin 80 75 70 65 60 55 50 amplitude (dB) Aliasing reduction butterfly effect for center bands Band 14 butterfly Band 14 Band 15 butterfly Band 15 Band 16 butterfly Band 16 Band 17 butterfly Band 17 38 / 71

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ΤΠϦΞε࡟ݮόλϑϥΠԋࢉ ▶ ৼ෯͕খ͘͞ ͳΔͱःஅಛ ੑ͸ѱԽ ▶ ՄௌҬଳΛ༏ ઌͨ݁͠Ռʁ 0 50 100 150 200 250 frequency bin 160 140 120 100 80 60 amplitude (dB) Aliasing reduction butterfly effect 39 / 71

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1. MP3 ֓ཁ MPEG/Audio ͷྺ࢙ ίʔσοΫߏ଄ 2. ϋΠϒϦουϑΟϧλόϯΫ ϑΟϧλόϯΫ MDCT (Modiﬁed Discrete Cosine Transform) ΤΠϦΞε࡟ݮόλϑϥΠԋࢉ 3. ྔࢠԽ 4. ූ߸Խ 5. ௌ֮৺ཧϞσϧ II 6. dist10 Τϯίʔμͷ֎෦ɾ಺෦ϧʔϓ 40 / 71

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ඇઢܗྔࢠԽ ϒϩοΫ௕͕ 36(long, start, stop) ͷͱ͖ɼྔࢠԽεϖ Ϋτϧ Xq i ɼٯྔࢠԽεϖΫτϧ ˆ Xi ͸ ∗2 Xq i = round sign(Xi ) Xi (Gl i )−1 3 4 (23) ˆ Xi = sign(Xq i )|Xq i |4 3 Gl i (24) Gl i = 21 4 g2−1 2 (1+scale)(sli +pi ) (25) ▶ gɿάϩʔόϧήΠϯ ▶ scaleɿεέʔϧϑΝΫλͷεέʔϧʢdist10 ΤϯίʔμʔͰ͸ৗʹ 0ʣ ▶ sli ɿεέʔϧϑΝΫλ ▶ pi ɿϓϦΤϯϑΝγε૿෯஋ʢdist10 ΤϯίʔμʔͰ͸ৗʹ 0ʣ ∗2round Λআ͚͹ɼٯྔࢠʹΑΓݩʹ໭Δ ˆ Xi = sign(Xi )|Xq i |4 3 Gl i = sign(Xi ) |Xi (Gl i )−1|3 4 4 3 Gl i = Xi (Gl i )−1Gl i = Xi 41 / 71

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ඇઢܗྔࢠԽ ϒϩοΫ௕͕ 12(short) ͷͱ͖ɼ Xq i = round sign(Xi ) Xi (Gs i )−1 3 4 (26) ˆ Xi = sign(Xq i )|Xq i |4 3 Gs i (27) Gs i = 21 4 g22sbgainb2−1 2 (1+scale)ssi (28) ▶ sbgainb ɿαϒϒϩοΫͷήΠϯʢdist10 ΤϯίʔμʔͰ͸ৗʹ 0ʣ ▶ ssi ɿγϣʔτϒϩοΫͷεέʔϧϑΝΫλ 42 / 71

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3/4৐ͷޮՌ x, x3 4 ͱͦΕΒͷඍ෼ 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 x x^(3/4) derivertive x derivertive x^(3/4) ▶ x3 4 ͸ 0 ۙ๣Ͱߴײ౓ʢlog ʹ͍ۙʣ ▶ 0 ۙ๣͸ࡉ͔͘ɼ≈ 0.3 Ҏ্͸ߥ͘ྔࢠԽ 43 / 71

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1. MP3 ֓ཁ MPEG/Audio ͷྺ࢙ ίʔσοΫߏ଄ 2. ϋΠϒϦουϑΟϧλόϯΫ ϑΟϧλόϯΫ MDCT (Modiﬁed Discrete Cosine Transform) ΤΠϦΞε࡟ݮόλϑϥΠԋࢉ 3. ྔࢠԽ 4. ූ߸Խ 5. ௌ֮৺ཧϞσϧ II 6. dist10 Τϯίʔμͷ֎෦ɾ಺෦ϧʔϓ 44 / 71

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MP3ͷූ߸Խ ྔࢠԽεϖΫτϧΛ۠෼ʹ෼͚ͯූ߸Խ big value େ͖͍஋͸ઢܗྔࢠԽΛ݉༻ ▶ region0,1,2 ͰҟͳΔϋϑϚϯςʔϒϧ Λ࢖༻ count1 data {−1, 0, 1} ͷΈͰූ߸Խ ▶ 1 ͭͷϋϑϚϯςʔϒϧΛ࢖༻ zero 0 ͷΈʢූ߸Խ͠ͳ͍ʣ 45 / 71

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MP3ͷූ߸Խʢৄࡉʣ big value ͷූ߸Խ ▶ ਺஋ 2 ͭ૊ Xq i , Xq i+1 Λ X = 16|Xq i | + |Xq i+1 | (29) ͱͯ͠ූ߸Խɽඇ 0 ͷ৔߹ʹූ߸ bit Λ෇Ճ ▶ ઢܗූ߸Խ͢Δ͔ͷ͖͍͠஋͸ςʔϒϧຖʹઃఆ count1 data ͷූ߸Խ ▶ ਺஋ 4 ͭ૊ Xq i , Xq i+1 , Xq i+2 , Xq i+3 Λ X = 8s(Xq i+2 ) + 4s(Xq i+3 ) + 2s(Xq i ) + s(Xq i+1 ) (30) ͱͯ͠ූ߸Խʢs(x)ɿx ̸= 0 ͳΒ 1ɼx = 0 ͳΒ 0ʣ ɽ ඇ 0 ͷ৔߹ʹූ߸ bit Λ෇Ճ 46 / 71

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1. MP3 ֓ཁ MPEG/Audio ͷྺ࢙ ίʔσοΫߏ଄ 2. ϋΠϒϦουϑΟϧλόϯΫ ϑΟϧλόϯΫ MDCT (Modiﬁed Discrete Cosine Transform) ΤΠϦΞε࡟ݮόλϑϥΠԋࢉ 3. ྔࢠԽ 4. ූ߸Խ 5. ௌ֮৺ཧϞσϧ II 6. dist10 Τϯίʔμͷ֎෦ɾ಺෦ϧʔϓ 47 / 71

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ௌ֮৺ཧϞσϧIIͷ֓ཁ ▶ LayerIII ͷௌ֮৺ཧϞσϧʢLayerI, II ͱ͸ҟͳΔʣ ▶ ग़ྗɿ৴߸ରϚεΫൺ (SMR∗3)ɼϒϩοΫλΠϓ ▶ ॲཧखॱ 1. ϑϨʔϜ੾Γग़͠ɾ૭͔͚ɾFFT 2. Unpredictability ܭࢉ 3. ύʔςΟγϣϯ͝ͱͷΤωϧΪʔܭࢉ 4. ޿͕Γؔ਺ (Spreading function) ͷ৞ΈࠐΈ 5. ϊΠζڐ༰Ϩϕϧܭࢉ 6. ௌ͖͍֮͠஋ܭࢉ 7. ஌֮Τϯτϩϐʔ (Psychoacoustic entropy) ܭࢉɾϒ ϩοΫλΠϓ൑ఆ 8. ৴߸ରϚεΫൺ (SMR) ܭࢉ ∗3Signal-to-Masking Ratio 48 / 71

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ϑϨʔϜ੾Γग़͠ɾ૭͔͚ɾFFT ▶ long ͱ short ͷ 2 ͭΛܭࢉ ▶ long ͸αΠζ 1024 ▶ short ͸αΠζ 256 Λ 3 ͭɽ৴߸ s[n] ͷ s[128b + n] b = 1, 2, 3 Λ࢖༻ ▶ Hanning ૭Λద༻ ▶ εϥΠυ (hop) αΠζ͸ 576ʢ=DCT εϖΫτϧ αΠζʣ FFT ܎਺Λ wl(long)ɼwsb(short, b = 1, 2, 3) ͱදه 49 / 71

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ϑϨʔϜ੾Γग़͠ɾ૭͔͚ɾFFT 102 103 104 Frequency (Hz) 20 0 20 40 60 80 100 120 Power (dB) energy ϐΞϊ (F0 =220Hz) ͷ long ͷΤωϧΪʔʢύϫʔεϖΫτϧʣ 50 / 71

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Unpredictabilityܭࢉ Unpredictability cw = ༧ଌͣ͠Β͞ͷई౓ cw[n] =      |wl[n]−wl∗[n]| |wl[n]|+|wl∗[n]| 0 ≤ n ≤ 5 |ws2 [j]−ws∗[j]| |ws2 [j]|+|ws∗[j]| j = ⌊(n + 2)/4⌋, 6 ≤ n ≤ 205 0.4 206 ≤ n ≤ 512 (31) ▶ wl∗, ws∗ɿৼ෯ͱҐ૬Λ௚ઢ༧ଌͨ͠εϖΫτϧ ▶ ௚ઢ༧ଌͷࣜʢwl′ ͸લϑϨʔϜͷ wlʣ ɿ wl∗[n] = (2|wl′[n]| − |wl′′[n]|) exp[j(2 arg(wl′[n]) − arg(wl′′[n]))] (32) ws∗[n] = (2|ws1 [n]| − |ws3 [n]|) exp[j(2 arg(ws1 [n]) − arg(ws3 [n]))] (33) ▶ ༧ଌ͕౰ͨΕ͹ cw[n] = 0ɼ֎ΕΔͱ cw[n] = 1 51 / 71

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ύʔςΟγϣϯ͝ͱͷΤωϧΪʔܭࢉ प೾਺ϏϯΛ”ύʔςΟγϣϯ (partition)”୯Ґʹ෼ׂ ▶ ebl, ebsɿύʔςΟγϣϯ಺ͷΤωϧΪʔΛ߹ࢉ ▶ cbɿUnpredictability ͰॏΈ͚ͮͨ͠ΤωϧΪʔΛ ߹ࢉ long, short ͷύʔςΟγϣϯ b Λ Pl b , Ps b ͱॻ͘ͱɼ ebl[b] = n∈Pl b |wl[n]|2 (34) cb[b] = n∈min Pl b cw[n]|wl[n]|2 (35) ebs[b] = n∈min Ps b |ws[n]|2 (36) 52 / 71

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ύʔςΟγϣϯ͝ͱͷΤωϧΪʔܭࢉ 0 10 20 30 40 50 60 Partition number 0 2500 5000 7500 10000 12500 15000 17500 20000 Frequency (Hz) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 min partition frequency max partition frequency αϯϓϦϯάप೾਺ 44.1kHz ͷ long ύʔςΟγϣϯɽBark εέʔ ϧࡉ෼ԽͱΈͳͤΔɽ਺஋ϥϕϧ͸εέʔϧϑΝΫλ൪߸ɽ 53 / 71

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ύʔςΟγϣϯ͝ͱͷΤωϧΪʔܭࢉ 102 103 104 Frequency (Hz) 20 0 20 40 60 80 100 120 Power (dB) energy partitoned energy ϐΞϊ (F0 =220Hz) ͷ long ύʔςΟγϣϯ෼ׂޙͷΤωϧΪʔ ebl 54 / 71

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޿͕Γؔ਺ͷ৞ΈࠐΈ ΤωϧΪʔʹ޿͕Γؔ਺ (Spreading function)SFl b , SFs b Λ৞ΈࠐΈɼௐ೾੒෼ͷϚεΩϯάΛߟྀ ▶ ecbl, ecbsɿSFl b , SFs b Λ৞Έ͜Μͩ ebl, ebs ▶ ctbɿSFl b Λ৞Έ͜Μͩ cb ecbl[b] = k SFl b [k]ebl[k] (37) ctb[b] = k SFl b [k]cb[k] (38) ecbs[b] = k SFs b [k]ebs[k] (39) 55 / 71

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޿͕Γؔ਺ͷ৞ΈࠐΈ 102 103 104 Frequency (Hz) 20 0 20 40 60 80 100 120 Power (dB) energy partitoned energy convolved partitoned energy ϐΞϊ (F0 =220Hz) ͷ޿͕Γؔ਺Λ৞Έ͜ΜͩΤωϧΪʔ ecbl 56 / 71

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Schroederͷ޿͕Γؔ਺SFschroeder [7] όʔΫεέʔϧʹม׵ͨ͠प೾਺ͷࠩ ∆z = z(fϚεΩʔ ) − z(fϚεΧʔ ) Λ༻͍ͯɼ 10 log10 SFschroeder (∆z) = 15.81 + 7.5(∆z + 0.474) − 17.5 1 + (∆z + 0.474)2 (40) (40) ࣜͰٻ·Δ஋͸ dB εέʔϧɽ1010 log10 SFschroeder(∆z) 10 Ͱৼ෯εέʔϧʹม׵ 57 / 71

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MP3ͷ޿͕Γؔ਺SF Schroeder ͷ޿͕Γؔ਺Λमਖ਼ tx = 3∆z ∆z ≤ 0 1.5∆z ∆z > 0 (41) x = 8 min{(tx − 0.5)2 − 2(tx − 0.5), 0} (42) y = 15.811389 + 7.5(tx + 0.474) − 17.5 1 + (tx + 0.474)2 (43) ͱͯ͠ɼSF ΛҎԼͰܭࢉɽ(41) ࣜΑΓɼ௿Ҭଆͷݮ ਰ͕ΑΓૣ͍ ∗4 10 log10 SF(∆z) = x + y y ≥ −60 0 otherwise (44) ∗4ύʔςΟγϣϯ͸εέʔϧϑΝΫλͷ໿ 3 ഒ෼ׂ͕ࡉ͔͍ͨΊʁ 58 / 71

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޿͕Γؔ਺ͷ֓ܗ 0 5 10 15 20 25 Bark scale 70 60 50 40 30 20 10 0 Gain (dB) MP3 bark:4.28 Schroeder bark:4.28 MP3 bark:9.21 Schroeder bark:9.21 MP3 bark:13.28 Schroeder bark:13.28 ͍͔ͭ͘ͷόʔΫ஋Ͱͷ޿͕Γؔ਺ɽഁઢ͸ Schroeder[7]ɽMP3 Ͱ͸-60dB ҎԼͷεέʔϧ͸ 0 ʹؙΊࠐΉͨΊઢ్͕੾Ε͍ͯΔɽ 59 / 71

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ϊΠζڐ༰Ϩϕϧܭࢉ ύʔςΟγϣϯ b ͷྔࢠԽϊΠζڐ༰ϨϕϧΛܭࢉ ∗5 tbb ܭࢉ tbb = 0 ͸ϊΠζɼtbb = 1 ͸ௐ೾੒෼Λࣔ͢ई౓ cbb = log max ctb[b] ecbl[b] , 0.01 (45) tbb = min{1.0, max{0.0, −0.299 − 0.43cbb}} (46) SNR ܭࢉ ϚεΫͷՃॏ࿨Ͱ SNR(dB) Λܭࢉɽ29.0 ͸ௐ೾ʹΑ ΔϊΠζͷϚεΫɼ6.0 ͸ϊΠζʹΑΔௐ೾ͷϚεΫ snr = max {minvalb , 29.0tbb + 6.0(1 − tbb)} (47) ڐ༰Ϩϕϧܭࢉ SNR ʹΤωϧΪʔΛ৐ͯ͡ڐ༰Ϩϕϧ nbl Λಘ Δɽ޿͕Γؔ਺Ͱ૿͑ͨΤωϧΪʔΛ໭͢ nbl[b] = ecbl[b] k SFb [k] × 10− snr 10 (48) ∗5short ͸লུɽSNR ΛςʔϒϧҾ͖ͯ͠ܭࢉ͢ΔҎ֎ಉ༷ͷͨΊ 60 / 71

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ϊΠζڐ༰Ϩϕϧܭࢉ 102 103 104 Frequency (Hz) 20 0 20 40 60 80 100 120 Power (dB) energy partitoned energy convolved partitoned energy noise permissive level ϐΞϊ (F0 =220Hz) ͷϊΠζڐ༰Ϩϕϧ nbl 61 / 71

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ௌ͖͍֮͠஋ܭࢉ લϑϨʔϜͷϊΠζڐ༰Ϩϕϧͱ࠷খՄௌύϫʔΛߟ ྀ͠ɼ͜ΕΛௌ͖͍֮͠஋ thrl ͱ͢Δ thrl[b] = max qthrb , min{2nbl′[b], 16nbl′′[b], nbl[b]} (49) ▶ qthrb ɿ࠷খՄௌύϫʔ ▶ nbl′, nbl′′ɿલͱ͞ΒʹͦͷલͷϊΠζڐ༰Ϩϕϧ ▶ લϑϨʔϜͷϊΠζΛ࢒͢ɽϓϦΤίʔରࡦ 62 / 71

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ௌ͖͍֮͠஋ܭࢉ 102 103 104 Frequency (Hz) 20 0 20 40 60 80 100 120 Power (dB) energy partitoned energy convolved partitoned energy noise permissive level threshold ϐΞϊ (F0 =220Hz) ͷௌ͖͍֮͠஋ thrl 63 / 71

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஌֮Τϯτϩϐʔܭࢉ ஌֮Τϯτϩϐʔ PE ΛҎԼͰܭࢉ PE = b |Pl b | log ebl[b] + 1 thrl[b] (50) ϑϨʔϜͷූ߸ԽʹඞཁͳϏοτ਺ͷ໨҆ ∗6 ▶ PE ≥ 1800 ͳΒ͹ short ϒϩοΫͱ൑ఆɽ͞΋ͳ ͘͹ long ϒϩοΫͱ൑ఆ ∗7 ∗6ಋग़͸ิ଍ 3 ࢀর ∗7dist10 Ͱ͸ҎલͷϒϩοΫ൑ఆ݁ՌΛอ࣋ɽ൑ఆͱϒϩοΫͷঢ়ଶભҠʹ ै͍݁ՌΛ start,stop ʹมߋ 64 / 71

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஌֮Τϯτϩϐʔܭࢉ 0 1 2 3 4 5 6 0.50 0.25 0.00 0.25 0.50 Amplitude Wave form 0 1 2 3 4 5 6 time (sec) 1000 2000 3000 4000 5000 Perceptual entropy Perceptual Entropy (PE) PE threshold ϐΞϊ (F0 =220Hz) ͷ೾ܗͱௌ֮Τϯτϩϐʔ PE ͷมԽ 65 / 71

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৴߸ରϚεΫൺ(SMR)ܭࢉ (ௌ͖͍֮͠஋)/(৴߸ύϫʔ) Λܭࢉɽಉ࣌ʹप೾਺෼ ׂΛεέʔϧϑΝΫλόϯυʹἧ͑Δɽ εέʔϧϑΝΫλόϯυ k ʹଐ͢ύʔςΟγϣϯͷू ߹Λ Pk ͱॻ͘ͱɼ en[k] = w1 [k]eb[min Pk ] + w2 [k]eb[max Pk ] + P ∈Pk b∈P eb[b] (51) thm[k] = w1 [k]thr[min Pk ] + w2 [k]thr[max Pk ] + P ∈Pk b∈P thr[b] (52) ͱͯ͠ʢw1 , w2 ͸ྡ઀όϯυؒͷॏΈఆ਺ ∗8ʣ ɼ ratio[k] = thm[k] en[k] en[k] > 0 0 otherwise (53) ∗8w1 [k] + w2 [k − 1] = 1 ͕੒ཱɽಋग़ํ๏͸ෆ໌... 66 / 71

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৴߸ରϚεΫൺ(SMR)ܭࢉ 102 103 104 Frequency (Hz) 40 30 20 10 0 Signal-to-Mask Ratio (SMR) (dB) ratio 40 50 60 70 80 90 100 110 Power (dB) partitoned energy threshold ϐΞϊ (F0 =220Hz) ͷ৴߸ରϚεΫൺ ratio 67 / 71

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1. MP3 ֓ཁ MPEG/Audio ͷྺ࢙ ίʔσοΫߏ଄ 2. ϋΠϒϦουϑΟϧλόϯΫ ϑΟϧλόϯΫ MDCT (Modiﬁed Discrete Cosine Transform) ΤΠϦΞε࡟ݮόλϑϥΠԋࢉ 3. ྔࢠԽ 4. ූ߸Խ 5. ௌ֮৺ཧϞσϧ II 6. dist10 Τϯίʔμͷ֎෦ɾ಺෦ϧʔϓ 68 / 71

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dist10ͷΤϯίʔυϧʔϓ ྔࢠԽήΠϯ (g, sli , ssi ) ͱϋϑ Ϛϯූ߸ςʔϒϧΛܾΊΔॲཧ ▶ ֎෦ϧʔϓʢྔࢠԽϊΠζ ੍ޚʣ ɾ಺෦ϧʔϓʢූ߸௕ ੍ޚʣͷ૬ޓ܁Γฦ͠ ▶ શεϖΫτϧ஋͕ 0 ͳΒ͹ εΩοϓ 69 / 71

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֎෦ϧʔϓ ྔࢠԽϊΠζ͕ௌ͖͍֮͠஋Λ ௒͑ͳ͍Α͏ʹ੍ޚ͢Δॲཧ ϓϦΤϯϑΝγε ߴҬόϯυ͕ ௌ͖͍֮͠஋Λ௒͑ ͍ͯͨΒɼ͖͍͠஋Λ ૿෯ ࿪Έ੍ޚ ௌ͖͍֮͠஋Λ௒͑ ͨεέʔϧϑΝΫλ όϯυͷྔࢠԽε ςοϓ෯Λݮগ 70 / 71

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಺෦ϧʔϓ ූ߸௕͕࢖༻ՄೳϏοτҎԼʹ ͳΔΑ͏ʹ੍ޚ͢Δॲཧ ྔࢠԽεςοϓ෯૿Ճ ྔࢠԽε ςοϓ෯Λ૿Ճʢά ϩʔόϧήΠϯ g ૿Ճ ʹ૬౰ʣ ූ߸௕ܭࢉ εϖΫτϧͷ۠෼ ׂɾςʔϒϧબ୒Λ ߦͬͯූ߸௕Λܭࢉ 71 / 71

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7. ࢀߟจݙ 8. ূ໌ ϑΟϧλόϯΫ MDCT ஌֮Τϯτϩϐʔಋग़ [19] 1 / 42

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ࢀߟจݙͷ঺հ ▶ [4]ɿPrincen–Bradley ͷ׬શ࠶ߏ੒৚݅ͷಋग़ ▶ [6]ɿMP3 ͷ֓ཁઆ໌ɽϔομɾαΠυΠϯϑΥϝʔγϣϯͷ಺༰ͷղઆ͸͔ͳΓৄ͍͠ ▶ [1]ɿMPEG/Audio ͷଞɼ2000 ೥୅લ൒ͷଞͷίʔσοΫͷ֓ཁΛղઆ ▶ [8]ɿMPEG ඪ४Խʹؔ͢Δॻ෺ɽٕज़͸֓ཁఔ౓ ▶ [2]ɿϚϧνϨʔτʹؔΘΔ৴߸ॲཧΛ޿ൣʹղઆ ▶ [3]ɿϑΟϧλόϯΫʹؔ͢Δৄࡉॻ ▶ [9]ɿDCT ʹؔ͢Δৄ͍͠ղઆ ▶ [10, 11, 12, 13, 14]ɿٕज़ղઆͱ؆қ MP3 ίʔσοΫͷ࣮૷ྫ ▶ [15]ɿMP3 ͷιʔε (dist10) ͷղઆɽͨͩ͠໦Λݟͯ৿Λݟͣͳҹ৅ɽ࣮૷ͷิ଍આ໌ ͱͯ͠͸༏ल͕ͩɼίʔυΛ਺ࣜʹ௚༁ͨ͠ॻ͖ํͷͨΊɼཧղͣ͠Β͍ɽ ▶ [16]ɿम࢜࿦จɽݚڀ͸ߴ଎Խ͕ͩɼMP3 ͷ֓ཁ͕·ͱ·͍ͬͯΔ ▶ [17]ɿ౦๺େҏ౻ڭतʹΑΔ MP3 ֓ཁղઆ ▶ [5, 18]ɿΤΠϦΞε࡟ݮͷ࿦จ ▶ [19]ɿϊΠζϚεΩϯάΛ༻͍ͨ஌֮Τϯτϩϐʔͷߟ͑ํʹ͍ͭͯͷݪ࿦จ ▶ [7]ɿϚεΩϯάۂઢͷϞσϧԽͱͦΕΛ༻͍ͨԻ੠ූ߸Խͷݪ࿦จ 2 / 42

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ࢀߟจݙ I [1] ౻ݪ༸. ը૾&Ի੠ѹॖٕज़ͷ͢΂ͯ : Πϯλʔωοτ/σΟδλϧςϨϏ/ϞόΠϧ௨ ৴࣌୅ͷඞਢٕज़. ୈ 6 ൛. Tech I. CQ ग़൛ࣾ, 2001. [2] وՈਔࢤ. ϚϧνϨʔτ৴߸ॲཧ. σΟδλϧ৴߸ॲཧγϦʔζ. তߊಊ, 1995. [3] P. P. Vaidyanathan et al. ϚϧνϨʔτ৴߸ॲཧͱϑΟϧλόϯΫ. σΟδλϧ৴߸ॲ ཧɾը૾ॲཧγϦʔζ. Պֶٕज़ग़൛, 2002. [4] John Princen and Alan Bradley. “Analysis/synthesis ﬁlter bank design based on time domain aliasing cancellation”. In: IEEE Transactions on Acoustics, Speech, and Signal Processing 34.5 (1986), pp. 1153–1161. [5] Bernd Edler. “Aliasing reduction in sub-bands of cascaded ﬁlter banks with decimation”. In: Electronics Letters 12.28 (1992), pp. 1104–1106. [6] Rassol Raissi. “The theory behind MP3”. In: MP3’ Tech (2002). [7] Manfred R Schroeder, Bishnu S Atal, and JL Hall. “Optimizing digital speech coders by exploiting masking properties of the human ear”. In: The Journal of the Acoustical Society of America 66.6 (1979), pp. 1647–1652. [8] ҆ాߒ. MPEG/ϚϧνϝσΟΞූ߸Խͷࠃࡍඪ४. ؙળ, 1994. 3 / 42

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ࢀߟจݙ II [9] وՈਔࢤ and ଜদਖ਼ޗ. ϚϧνϝσΟΞٕज़ͷجૅ DCT(཭ࢄίαΠϯม׵) ೖ໳ : JPEG/MPEG ͔Β΢ΣʔϒϨοτ, ॏෳ௚ަม׵ (LOT) ·Ͱ. I/F essence. CQ ग़൛, 1997. [10] খਿಞ࢙. Interface Aug.2001 ୈ 1 ճԻڹѹॖٕज़ͷجૅ MP3 ͱ౳ՁతͳγεςϜͷ ߏஙͱαϒόϯυϑΟϧλόϯΫͷઃܭ. CQ ग़൛ࣾ, 2001. [11] খਿಞ࢙ and ৓Լ૱. Interface Sep.2001 ୈ 2 ճԻڹѹॖٕज़ͷجૅ MP3 ͱ౳Ձతͳ γεςϜͷߏஙͱ MDCT ͷઃܭ. CQ ग़൛ࣾ, 2001. [12] খਿಞ࢙ and ৓Լ૱. Interface Nov.2001 ୈ 3 ճԻڹѹॖٕज़ͷجૅ MP3 ͱ౳Ձతͳ γεςϜͷߏஙͱϋΠϒϦουϑΟϧλόϯΫͷઃܭ. CQ ग़൛ࣾ, 2001. [13] খਿಞ࢙ and ৓Լ૱. Interface Jan.2002 ୈ 4 ճԻڹѹॖٕज़ͷجૅ MP3 ͱ౳Ձతͳ γεςϜͷߏஙͱඇઢܗྔࢠԽث/ූ߸Խثͷઃܭ. CQ ग़൛ࣾ, 2002. [14] খਿಞ࢙ and ৓Լ૱. Interface Feb.2002 ୈ 4 ճԻڹѹॖٕज़ͷجૅ MP3 ͱ౳Ձతͳ γεςϜͷߏஙͱƂͦͷͨΊͷϏοτετϦʔϜͷઃܭ. CQ ग़൛ࣾ, 2002. [15] Ӝాහಓ. ৄࡉ MP3 ϚχϡΞϧ. ΤϜݚ, 1999. [16] ཥࣳ߶. MPEG Audio Layer3 ͷΤϯίʔμͷߴ଎Խॲཧʹؔ͢Δݚڀ. http://labo.nshimizu.com/thesis/master/1aepm054.pdf. [Online; accessed 28-Jul-2024]. 2002. 4 / 42

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ࢀߟจݙ III [17] ҏ౻জଇ. ߴޮ཰Իָූ߸Խ ʕMP3 ৄղʕ. https://www.slideshare.net/akinoriito549/slides-43584939. [Online; accessed 28-Jul-2024]. 2015. [18] Chi-Min Liu and Wen-Chieh Lee. “The design of a hybrid ﬁlter bank for the psychoacoustic model in ISO/MPEG phases 1, 2 audio encoder”. In: IEEE transactions on consumer electronics 43.3 (1997), pp. 586–592. [19] James D Johnston. “Estimation of perceptual entropy using noise masking criteria”. In: Icassp-88., international conference on acoustics, speech, and signal processing. IEEE. 1988, pp. 2524–2527. 5 / 42

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7. ࢀߟจݙ 8. ূ໌ ϑΟϧλόϯΫ MDCT ஌֮Τϯτϩϐʔಋग़ [19] 6 / 42

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׬શ࠶ߏ੒ ஗Ԇɾఆ਺ഒΛআ͖ೖग़ྗ͕Ұக͢Δ͜ͱɿ ˆ x[n] = cx[n − n0 ], c ̸= 0 (54) ͜Ε͸ z ྖҬͰɼ ˆ X(z) = cz−n0X(z) (55) ͱͳΔ͜ͱͱ౳Ձ 7 / 42

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ϙϦϑΣʔζදݱ Hk (z) ͷΠϯύϧεԠ౴Λ hk [n] ͱॻ͘ͱ͖ɼ Hk (z) = ∞ n=−∞ hk [n]z−n = ∞ n=−∞ M−1 l=0 hk [nM + l]z−(nM+l) = M−1 l=0 z−l ∞ n=−∞ hk [nM + l]z−nM = M−1 l=0 Ek,l (zM )z−l hk ͷʢλΠϓ IʣϙϦϑΣʔζදݱ Ek,l (z) = ∞ n=−∞ hk [nM + l]z−n (56) 8 / 42

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ϙϦϑΣʔζදݱ Fk (z) ͷΠϯύϧεԠ౴Λ fk [n] ͱॻ͘ͱ͖ɼ Fk(z) = ∞ n=−∞ fk[n]z−n = ∞ n=−∞ M−1 l=0 fk[nM + l]z−(nM+l) = ∞ n=−∞ M−1 l′=0 fk[nM + M − 1 − l′]z−(nM+M−1−l′) = M−1 l′=0 z−(M−1−l′) ∞ n=−∞ fk[nM + M − 1 − l′]z−nM = M−1 l′=0 z−(M−1−l′)Rk,l(zM ) fk ͷʢλΠϓ IIʣϙϦϑΣʔζදݱ Rk,l (z) = ∞ n=−∞ fk [nM + M − 1 − l]z−n (57) 9 / 42

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ϙϦϑΣʔζߦྻදݱ (56) ࣜΛ l ʹ͍ͭͯฒ΂ɼߦྻදݱ͢Δͱ     H0(z) H1(z) HM−1(z)     h(z) =     E0,0(zM ) E0,1(zM ) E0,M−1(zM ) E1,0(zM ) E1,1(zM ) E1,M−1(zM ) EM−1,0(zM ) EM−1,1(zM ) EM−1,M−1(zM )     E(z)     1 z−1 z−(M−1)     e(z) (57) ࣜ΋ಉ༷ʹͯ͠ɼҎԼͷΑ͏ʹॻ͚Δ     F0(z) F1(z) FM−1(z)     f(z) =     R0,0(zM ) R0,1(zM ) R0,M−1(zM ) R1,0(zM ) R1,1(zM ) R1,M−1(zM ) RM−1,0(zM ) RM−1,1(zM ) RM−1,M−1(zM )     T R(z)T     z−(M−1) z−(M−1)+1 1     z−(M−1)e(z−1) 10 / 42

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ϙϦϑΣʔζߦྻදݱ ˜ e(z) = e(z−1)T ͱ͢Δͱߦྻදݱ͸ h(z) = E(z)e(z) (58) f(z)T = z−(M−1)˜ e(z)R(z) (59) ͱ·ͱΊΒΕΔɽE(z), R(z) ΛϙϦϑΣʔζߦྻͱ ͍͏ 11 / 42

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ϙϦϑΣʔζߦྻදݱ E(z), R(z) ʹΑΓɼΞφϥΠβɾγϯηαΠβ͸ҎԼ ͷΑ͏ʹมܗͰ͖Δ 12 / 42

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ϙϦϑΣʔζߦྻදݱ E(z), R(z) ʹΑΓɼM ෼ׂϑΟϧλόϯΫ͸ҎԼͷΑ ͏ʹදͤΔ 13 / 42

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׬શ࠶ߏ੒M ෼ׂϑΟϧλόϯΫ R(z)E(z) = az−m0I (a ̸= 0, m0 ∈ N) (60) ͳΒ͹ɼM ෼ׂϑΟϧλόϯΫ͸׬શ࠶ߏ੒ a a∵ ֤όϯυͷ஗Ԇ͕ K ͳΒ͹ɼ ˆ X(z) = aMz−(M−1+K)X(z) R(z)E(z) = az−m0 I Λຬͨ͢ M ෼ׂϑΟϧλόϯΫ 14 / 42

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׬શ࠶ߏ੒M ෼ׂϑΟϧλόϯΫ (60)ࣜΑΓɼR(z) = az−m0E(z)−1 ͳΒ͹׬શ࠶ߏ੒ɽ ▶ ͔͠͠ɼE(z)−1 ͷܭࢉʹ໰୊ΛሃΉɽ ୅ΘΓʹɼE(z) ͕ύϥϢχλϦ ∗9 E(z)E(z) = dI, d ̸= 0 (61) ͳΒ͹ɼ R(z) = az−m0E(z) ͱ͢ΔͱϑΟϧλόϯΫ͸׬શ࠶ߏ੒ɽ ∗9E(z) = E∗ (z−1) ͰɼԼ෇͖ͷ ∗ ͸܎਺ͷෳૉڞ໾ 15 / 42

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ίαΠϯมௐϑΟϧλόϯΫ ࣜ (2) ΑΓɼ෼ੳ߹੒ϑΟϧλ Fk (z)Hk (z) ͸௚ઢҐ૬ ಛੑΛ΋ͭ ʢূ໌ʣ Fk(z) = ∞ n=−∞ fk[n]z−n = ∞ n=−∞ hk[L − 1 − n]z−n = ∞ n′=−∞ hk[n′]z−(L−1−n′) = z−(L−1) ∞ n′=−∞ hk[n′]zn′ = z−(L−1)Hk(z−1) Hk(z) ͷप೾਺ಛੑΛʢۃ࠲ඪͰʣHk(ω) = |Hk(ω)| exp[jψ(ω)] ͱॻ͘ͱɼ Fk(ω) = exp[−j(L − 1)ω]Hk(−ω) = exp[−j(L − 1)ω]|Hk(−ω)| exp[−jψ(ω)] = exp[−j(L − 1)ω]|Hk(ω)| exp[−jψ(ω)] ʢ∵ ࣮܎਺ FIR ͷৼ෯ಛੑ͸ۮʣ ͔ͩΒɼFk(ω)Hk(ω) = exp[−j(L − 1)ω]|Hk(ω)|2 ͱͳͬͯ௚ઢҐ૬ಛੑΛ΋ͭɽ 16 / 42

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ίαΠϯมௐϑΟϧλόϯΫ hk [n] ͷ఻ୡؔ਺Λมܗ͢ΔɽճసҼࢠ W2M ∗10 ΑΓ cos π M k + 1 2 n − L − 1 2 + θk = 1 2 exp j π M k + 1 2 n − L − 1 2 + θk + exp −j π M k + 1 2 n − L − 1 2 + θk = 1 2 exp(jθk)W (k+ 1 2 ) L−1 2 2M W−(k+ 1 2 )n 2M + exp(−jθk)W−(k+ 1 2 ) L−1 2 2M W (k+ 1 2 )n 2M ͜ΕΛ (1) ࣜʹ୅ೖ͢Δͱɼ Hk(z) = exp(jθk)W (k+ 1 2 ) L−1 2 2M ∞ n=−∞ p0[n]W−(k+ 1 2 )n 2M z−n + exp(−jθk)W−(k+ 1 2 ) L−1 2 2M ∞ n=−∞ p0[n]W (k+ 1 2 )n 2M z−n = exp(jθk)W (k+ 1 2 ) L−1 2 2M P0 Wk+ 1 2 2M z + exp(−jθk)W−(k+ 1 2 ) L−1 2 2M P0 W−(k+ 1 2 ) 2M z ∗10W2M = exp −j 2π 2M = exp −j π M 17 / 42

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ίαΠϯมௐϑΟϧλόϯΫ ͞Βʹ (69) ࣜΛ୅ೖ͢Δͱ ∗11 Hk(z) = exp(jθk)W (k+ 1 2 ) L−1 2 2M 2M−1 l=0 Wk+ 1 2 2M z −l Gl Wk+ 1 2 2M z 2M + exp(−jθk)W−(k+ 1 2 ) L−1 2 2M 2M−1 l=0 W−(k+ 1 2 ) 2M z −l Gl W−(k+ 1 2 ) 2M z 2M = exp(jθk)W (k+ 1 2 ) L−1 2 2M 2M−1 l=0 W−l(k+ 1 2 ) 2M z−lGl(−z2M ) + exp(−jθk)W−(k+ 1 2 ) L−1 2 2M 2M−1 l=0 Wl(k+ 1 2 ) 2M z−lGl(−z2M ) = 2M−1 l=0 exp(jθk)W (k+ 1 2 ) L−1 2 2M W−l(k+ 1 2 ) 2M + exp(−jθk)W−(k+ 1 2 ) L−1 2 2M Wl(k+ 1 2 ) 2M z−lGl(−z2M ) = 2M−1 l=0 2 cos π M k + 1 2 l − L − 1 2 + θk z−lGl(−z2M ) ∗11 W±(k+ 1 2 ) 2M 2M = exp ∓j π M k + 1 2 2M = exp[∓jπ(2k + 1)] = −1 Λ ࢖༻ 18 / 42

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ίαΠϯมௐϑΟϧλόϯΫ ͞Βʹมܗ͢Δͱ Hk (z) = M−1 l=0 z−l tk,l Gl (−z2M ) + z−M tk,M+l GM+l (−z2M ) (62) tk,l := 2 cos π M k + 1 2 l − L − 1 2 + θk (56) ࣜͱ (62) ࣜΛݟൺ΂Δͱɼ Ek,l (z) = tk,l Gl (−z2) + z−1tk,M+l GM+l (−z2) (63) 19 / 42

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ίαΠϯมௐϑΟϧλόϯΫ (63) ࣜΑΓɼΞφϥΠβͷϙϦϑΣʔζߦྻ͸ɼ E(z) =     t0,0 t0,2M−1 t1,0 t1,2M−1 tM−1,0 tM−1,2M−1     T         G0 (−z2) GM−1 (−z2) z−1GM (−z2) z−1G2M−1 (−z2)         = T G0 (z2) z−1G1 (z2) (64) Gi (z) := diag GMi (−z) GMi+1 (−z) GMi+M−1 (−z) (65) ͱॻ͚Δ 20 / 42

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ίαΠϯมௐϑΟϧλόϯΫ γϯηαΠβΛߏ੒͢ΔɽϑΟϧλ܎਺͸࣮͔ͩΒɼ E(z) = E∗ (z−1)T = G0 (z−1) zG1 (z−1) T T (66) Gl (z) ͷ࣍਺͸ 2K − 1 ͰɼҼՌੑͷͨΊʹ͸ 2K − 1 ͷ஗Ԇ͕͍ΔͨΊ R(z) = z−(2K−1)E(z) = z−(2K−1) G0 (z−1) zG1 (z−1) T T (67) 21 / 42

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ίαΠϯมௐϑΟϧλόϯΫ ΞφϥΠβͷߏ੒ 22 / 42

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ίαΠϯมௐϑΟϧλόϯΫ γϯηαΠβͷߏ੒ 23 / 42

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ίαΠϯมௐϑΟϧλόϯΫ ׬શ࠶ߏ੒৚݅Λಋ͘ɽ(64)ɼ(67) ࣜΑΓɼ R(z)E(z) = z−(2K−1)E(z)E(z) = z−(2K−1) G0 (z−1) zG1 (z−1) T TT G0 (z) z−1G1 (z) = 2Mz−(2K−1) G0 (z−1)G0 (z) + G1 (z−1)G1 (z) ͜͜ͰɼT TT = 2MIʢޙͰࣔ͢ʣ ɽG0 , G1 ͸ର֯ߦ ྻ͔ͩΒɼk = 0, ..., M − 1 ʹର͠ɼ Gk (z−1)Gk (z) + GM+k (z−1)GM+k (z) = α Λຬͨͤ͹׬શ࠶ߏ੒ͱͳΔ 24 / 42

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ίαΠϯมௐϑΟϧλόϯΫ ίαΠϯมௐϑΟϧλόϯΫͷ׬શ࠶ߏ੒৚݅ k = 0, ..., M − 1 ʹର͠ɼఆ਺ α ∈ R ͕͋ͬͯ Gk (z−1)Gk (z) + GM+k (z−1)GM+k (z) = α (68) ͱͳΔ͜ͱɽ͜͜Ͱɼ Gk (z) = ∞ n=−∞ p0 [2Mn + k]z−n (69) Gk (z) ͸ p0 [n] ͷϙϦϑΣʔζදݱ ຊ৚݅͸ిྗ૬ิ৚݅ͱ΋͍͏ 25 / 42

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T TT = 2MI ͷূ໌ T TT ij = M−1 k=0 tk,i tk,j Ͱ͋Γɼ tk,itk,j = 4 cos π M k + 1 2 i − L − 1 2 + (−1)k π 4 cos π M k + 1 2 j − L − 1 2 + (−1)k π 4 = 2 cos π M k + 1 2 {i + j − (L − 1)} + (−1)k π 2 + cos π M k + 1 2 (i − j) i + j − (L − 1) = A ͱ͓͘ͱɼ cos π M k + 1 2 A + (−1)k π 2 = cos π M k + 1 2 A cos (−1)k π 2 − sin π M k + 1 2 A sin (−1)k π 2 = −(−1)k sin π M k + 1 2 A 26 / 42

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T TT = 2MI ͷূ໌ ͜͜Ͱɼ M−1 k=0 cos π M k + 1 2 (i − j) = Mδij (70) M−1 k=0 (−1)k sin π M k + 1 2 A = 0 (71) ʢδij ɿΫϩωοΧʔͷσϧλʣΛࣔͤ͹ɼ T TT ij = 2Mδij ͱͳΓ໋୊͕ࣔͤΔɽ࣍ϖʔδ͔Βܭࢉ݁ՌΛࡌͤΔ 27 / 42

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T TT = 2MI ͷূ໌ (70) ࣜΛࣔ͢ɽi = j ͷͱ͖ɼ M−1 k=0 cos π M k + 1 2 (i − j) = Mɽ i ̸= j ͷͱ͖ɼ M−1 k=0 cos π M k + 1 2 (i − j) = 1 2 M−1 k=0 exp j π M k + 1 2 (i − j) + exp −j π M k + 1 2 (i − j) = 1 2 M−1 k=0 W−(k+ 1 2 )(i−j) 2M + W (k+ 1 2 )(i−j) 2M = 1 2 W− i−j 2 2M M−1 k=0 W−(i−j)k 2M + W i−j 2 2M M−1 k=0 W(i−j)k 2M = 1 2   W− i−j 2 2M W−(i−j) 2M − 1 (−1)−(i−j) − 1 + W i−j 2 2M Wi−j 2M − 1 (−1)i−j − 1   ࠷ޙͷࣜมܗͰ౳ൺڃ਺ͷ࿨ M−1 k=0 W(i−j)k 2M = W (i−j)M 2M −1 W i−j 2M −1 = (−1)i−j −1 W i−j 2M −1 Λ࢖༻ 28 / 42

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T TT = 2MI ͷূ໌ ͞Βʹࣜมܗ͢Δͱɼ M−1 k=0 cos π M k + 1 2 (i − j) = 1 2   W− i−j 2 2M W−(i−j) 2M − 1 (−1)−(i−j) − 1 + W i−j 2 2M Wi−j 2M − 1 (−1)i−j − 1   = W− i−j 2 2M (Wi−j 2M − 1) (−1)−(i−j) − 1 + W i−j 2 2M (W−(i−j) 2M − 1) (−1)i−j − 1 2(W−(i−j) 2M − 1)(Wi−j 2M − 1) = (W i−j 2 2M − W− i−j 2 2M ) (−1)−(i−j) − 1 + (W− i−j 2 2M − W i−j 2 2M ) (−1)i−j − 1 2(W−(i−j) 2M − 1)(Wi−j 2M − 1) = (W i−j 2 2M − W− i−j 2 2M ) (−1)i−j − 1 + (W− i−j 2 2M − W i−j 2 2M ) (−1)i−j − 1 2(W−(i−j) 2M − 1)(Wi−j 2M − 1) = 0 Αͬͯ (70) ͕ࣜࣔ͞Εͨ 29 / 42

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T TT = 2MI ͷূ໌ (71) ࣜΛࣔ͢ɽ M−1 k=0 (−1)k sin π M k + 1 2 A = 1 j2 M−1 k=0 (−1)k exp j π M k + 1 2 A − exp −j π M k + 1 2 A = 1 j2 M−1 k=0 (−1)k W−A(k+ 1 2 ) 2M − WA(k+ 1 2 ) 2M = 1 j2 W− A 2 2M M−1 k=0 (−1)kW−Ak 2M − W A 2 2M M−1 k=0 (−1)kWAk 2M = 1 j2 W− A 2 2M M−1 k=0 W−(M+A)k 2M − W A 2 2M M−1 k=0 W(M+A)k 2M ʢ∵ (−1)k = WM 2M = W−M 2M ʣ = 1 j2 W− A 2 2M (−1)−(M+A) − 1 W−(M+A) 2M − 1 − W A 2 2M (−1)M+A − 1 WM+A 2M − 1 ʢ∵ ౳ൺڃ਺ͷ࿨ͷެࣜʣ 30 / 42

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T TT = 2MI ͷূ໌ ͞ΒʹܭࢉΛਐΊΔͱɼ M−1 k=0 (−1)k sin π M k + 1 2 A = W− A 2 2M (−1)−(M+A) − 1 (WM+A 2M − 1) − W A 2 2M (−1)M+A − 1 (W−(M+A) 2M − 1) j2(W−(M+A) 2M − 1)(W(M+A) 2M − 1) = (−1)M+A − 1 W− A 2 2M (WM+A 2M − 1) − W A 2 2M (W−(M+A) 2M − 1) j2(W−(M+A) 2M − 1)(W(M+A) 2M − 1) = (−1)M+A − 1 WM+ A 2 2M − W− A 2 2M − W−M− A 2 2M + W A 2 2M j2(W−(M+A) 2M − 1)(W(M+A) 2M − 1) = (−1)M+A − 1 −W A 2 2M − W− A 2 2M + W− A 2 2M + W A 2 2M j2(W−(M+A) 2M − 1)(W(M+A) 2M − 1) = 0 Αͬͯɼ(71) ͕ࣜࣔ͞Εͨɽ 31 / 42

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MP3ͷϑΟϧλόϯΫͷಛੑ 0 100 200 300 400 500 index 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 amplitude MP3 Decoder prototype filter coefficients ▶ σίʔμͷ܎਺ɽΤϯίʔμͷ܎਺ͷ 32 ഒ 32 / 42

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ిྗ૬ิ৚݅ͷ֬ೝ MP3 ͷϑΟϧλόϯΫͰ͸ Gk(z) = ∞ n=−∞ p0[2nM + k]z−n = 7 n=0 p0[64n + k]z−n = 7 n=0 (−1)nC64+kz−n GM+k(z) = 7 n=0 p0[64n + 32 + k]z−n = 7 n=0 (−1)nC64n+32+kz−n Gk(z−1) = ∞ n=−∞ p0[64n + k](z−1)−n = ∞ n=−∞ p0[−64n + k]z−n = 0 n=−7 (−1)nC−64n+kz−n GM+k(z−1) = 7 n=0 p0[−64n + 32 + k]z−n = 7 n=0 (−1)nC−64n+32+kz−n k = 0, ..., 31 Ͱ࣮ࡍʹܭࢉ͢Δͱɼ Gk (z−1)Gk (z) + GM+k (z−1)GM+k (z) ≈ 1 512 33 / 42

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ిྗ૬ิ৚݅ͷ֬ೝʢܭࢉ݁Ռʣ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 normalized frequency 0.00000 0.00025 0.00050 0.00075 0.00100 0.00125 0.00150 0.00175 0.00200 amplitude Real part Imaginary part MP3 prototype filter power complementary condition check ”΄΅”׬શ࠶ߏ੒ͱݴͬͯΑ͍ 34 / 42

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MDCTɾIMDCTʹΑΔ࠶ߏ੒৴߸ ࣜ (10) ʹࣜ (9) Λ୅ೖ͢Δͱɼ y[n] = 2 N N−1 k=0 2N−1 m=0 x[m] cos π N k + 1 2 m + 1 2 + N 2 cos π N k + 1 2 n + 1 2 + N 2 = 1 N N−1 k=0 2N−1 m=0 x[m] cos π N k + 1 2 (n + m + 1 + N) + cos π N k + 1 2 (n − m) = 1 N 2N−1 m=0 x[m] N−1 k=0 cos π N k + 1 2 (n + m + 1 + N) + cos π N k + 1 2 (n − m) ͜͜Ͱɼ In := N−1 k=0 cos π N k + 1 2 n (72) Λܭࢉ͍ͯ͘͠ 35 / 42

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MDCTɾIMDCTʹΑΔ࠶ߏ੒৴߸ In = N−1 k=0 cos π N k + 1 2 n = N−1 k=0 cos π 2N (2k + 1)n = 1 2 N−1 k=0 exp j π 2N (2k + 1)n + exp −j π 2N (2k + 1)n = 1 2 N−1 k=0 W− 2k+1 2 n 2N + W 2k+1 2 n 2N = 1 2 W− n 2 2N N−1 k=0 W−nk 2N + W n 2 2N N−1 k=0 Wnk 2N ͜͜Ͱɼ N−1 k=0 Wnk 2N = N−1 k=0 (Wn 2N )k =            N−1 k=0 1k = N ʢn ͕ 2N ͷഒ਺ʣ 1 Wn 2N N − 1 Wn 2N − 1 = (−1)n − 1 Wn 2N − 1 ʢn ͕ 2N ͷഒ਺Ͱ͸ͳ͍ʣ ͔Βɼ৔߹෼͚ͯ͠ߟ͑Δ 36 / 42

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MDCTɾIMDCTʹΑΔ࠶ߏ੒৴߸ n ͕ 2N ͷഒ਺ͷͱ͖ɼn = 2Nm (m ∈ Z) ͱॻ͚ͯɼ In = N 2 W−2Nm 2N + W2Nm 2N = N 2 (−1)−m + (−1)m = N ʢmɿۮ਺ ⇐⇒ n = 0, ±4N, ±8Nʣ −N ʢmɿح਺ ⇐⇒ n = ±2N, ±6N, ±10Nʣ n ͕ 2N ͷഒ਺Ͱ͸ͳ͍ͱ͖ɼ In = 1 2 W− n 2 2N (−1)−n − 1 W−n 2N − 1 + W n 2 2N (−1)n − 1 Wn 2N − 1 = W− n 2 2N (−1)−n − 1 (Wn 2N − 1) + W n 2 2N {(−1)n − 1} (W−n 2N − 1) 2(W−l 2N − 1)(Wn 2N − 1) = W− n 2 2N (−1)−nWn 2N − (−1)−n − Wn 2N + 1 + W n 2 2N (−1)nW−n 2N − (−1)n − W−n 2N + 1 2(W−l 2N − 1)(Wn 2N − 1) = (−1)−nW n 2 2N − (−1)−nW− n 2 2N − W n 2 2N + W− n 2 2N + (−1)nW− n 2 2N − (−1)nW n 2 2N − W− n 2 2N + W n 2 2N 2(W−l 2N − 1)(Wn 2N − 1) = 0 37 / 42

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MDCTɾIMDCTʹΑΔ࠶ߏ੒৴߸ ·ͱΊΔͱɼ In =    N (n = 0, ±4N, ±8N, ...) −N (n = ±2N, ±6N, ±10N, ...) 0 (otherwise) (73) ͱͳΔɽࣜ (73) Λ࢖͑͹ɼ 2N−1 m=0 x[m]Im−n = x[n]I0 = Nx[n] (n = 0, ..., N − 1) x[n]I0 = Nx[n] (n = N, ..., 2N − 1) 2N−1 m=0 x[m]In+m+1+N = x[N − 1 − n]I2N = −Nx[N − 1 − n] (n = 0, ..., N − 1) x[3N − 1 − n]I4N = Nx[3N − 1 − n] (n = N, ..., 2N − 1) ͔ͩΒɼ y[n] = 1 N 2N−1 m=0 x[m](In+m+1+N + Im−n) = x[n] − x[N − 1 − n] (n = 0, ..., N − 1) x[n] + x[3N − 1 − n] (n = 0, ..., N − 1) ͱͳΔɽ 38 / 42

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஌֮Τϯτϩϐʔಋग़ ௌ͖͍֮͠஋ Tb ʹྔࢠԽ෼ࢄʢύϫʔʣΛ߹ΘͤΔ ▶ ֤प೾਺ϏϯΛྔࢠԽεςοϓ෯ ∆b ͰҰ༷ྔࢠ Խ ⇒ ྔࢠԽޡ͕ࠩҰ༷ʹੜى͢ΔͳΒɼྔࢠԽ ޡࠩ෼ࢄ͸Ϗϯ͋ͨΓ ∆2 b 12 ▶ ύʔςΟγϣϯ b ʹؚ·ΕΔϏϯ਺Λ nb ͱ͢Δ ͱɼϏϯ͋ͨΓͷௌ͖͍֮͠஋͸ Tb nb ͜ΕΒΛ౳͍͠ͱ͢Δͱɼ Tb nb = ∆2 b 12 =⇒ ∆b = 12Tb nb (74) 39 / 42

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஌֮Τϯτϩϐʔಋग़ εϖΫτϧͷූ߸ԽʹඞཁͳϏοτ਺Λಋग़ ▶ Ϗϯ͋ͨΓͷฏۉৼ෯εϖΫτϧ͸ Ib /nb ▶ εϖΫτϧ͸ [−⌈ Ib /nb ⌉, ⌈ Ib /nb ⌉] ͷൣғ ⇒ ූ߸Խൣғͷ෯͸ 2 Ib /nb ൣғΛεςοϓ෯ ∆b ͰׂΔͱූ߸ԽγϯϘϧݸ਺ʹ ͳΔɽ࣮ڏ྆࣠Ͱූ߸ԽʹඞཁͳϏοτ਺͸ log2 2 Ib /nb ∆b + log2 2 Ib /nb ∆b = 2 log2 2 Ib /nb ∆b (75) 40 / 42

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஌֮Τϯτϩϐʔಋग़ (75) ࣜΛมܗ 2 log2 2 Ib /nb ∆b = log2 4Ib nb ∆2 b = log2 Ib 3Tb ∝ log Ib 3Tb = log Ib Tb + const. શप೾਺Ϗϯͷූ߸ԽʹඞཁͳϏοτ਺͸ b nb log Ib Tb + const. (76) ʹൺྫ 41 / 42

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஌֮Τϯτϩϐʔಋग़ ن֨Ͱ͸஌֮Τϯτϩϐʔ PE Λ PE = b nn log Ib + 1 Tb (77) Ͱఆٛɽdist10 Ͱ͸ແԻྖҬͰ Tb = 0 ͱͳΔͨΊɼ PE = b nn log Ib + 1 Tb + 1 (78) Ͱܭࢉ ∗12 ∗12༗Ի۠ؒͰ Ib , Tb ͸େ͖͍ͷͰɼۙࣅͱͯ͠͸໰୊ͳ͍૝ఆ 42 / 42